To determine the diameter of a hemisphere when its total surface area is given as [tex]\(36 \pi \, \text{cm}^2\)[/tex], we need to follow a series of steps involving the formula for the surface area of the hemisphere and solving for the radius and then the diameter. Here's a detailed breakdown:
1. Understanding the Surface Area Formula:
The total surface area (TSA) of a hemisphere includes both its curved surface and its flat circular base. The formula for the total surface area of a hemisphere is:
[tex]\[
\text{TSA} = 3 \pi r^2
\][/tex]
where [tex]\(r\)[/tex] is the radius of the hemisphere.
2. Given Total Surface Area:
The problem states the total surface area is [tex]\(36 \pi \, \text{cm}^2\)[/tex]. This means:
[tex]\[
3 \pi r^2 = 36 \pi
\][/tex]
3. Isolate and Solve for [tex]\(r^2\)[/tex]:
To find [tex]\(r^2\)[/tex], we can divide both sides of the equation by [tex]\(\pi\)[/tex], and then by 3:
[tex]\[
3 r^2 = 36
\][/tex]
[tex]\[
r^2 = \frac{36}{3}
\][/tex]
[tex]\[
r^2 = 12
\][/tex]
4. Solve for the Radius [tex]\(r\)[/tex]:
Taking the square root of both sides to find [tex]\(r\)[/tex]:
[tex]\[
r = \sqrt{12}
\][/tex]
[tex]\[
r = 2\sqrt{3}
\][/tex]
This numerical value is:
[tex]\[
r \approx 3.464
\][/tex]
5. Calculate the Diameter:
The diameter [tex]\(d\)[/tex] of the hemisphere is twice the radius, given by:
[tex]\[
d = 2r
\][/tex]
Substituting the value of [tex]\(r\)[/tex]:
[tex]\[
d = 2 \times 3.464 \approx 6.928
\][/tex]
So, the diameter of the hemisphere is approximately [tex]\(6.928 \, \text{cm}\)[/tex].
